Question: Simplify $(1 + \tan 20^\circ)(1 + \tan 25^\circ).$
From the angle addition formula,
\[1 = \tan 45^\circ = \tan (20^\circ + 25^\circ) = \frac{\tan 20^\circ + \tan 25^\circ}{1 - \tan 20^\circ \tan 25^\circ},\]so $\tan 20^\circ + \tan 25^\circ = 1 - \tan 20^\circ \tan 25^\circ.$

Then
\[(1 + \tan 20^\circ)(1 + \tan 25^\circ) = 1 + \tan 20^\circ + \tan 25^\circ + \tan 20^\circ \tan 25^\circ = \boxed{2}.\]